Vehicles with an engine that automatically stops when vehicles are stopped during traveling and a prescribed engine stop condition is met to improve fuel economy, reduce exhaust emission, or reduce noise have been proposed and brought into practical use. Meanwhile, a coast-stop vehicle capable of further reducing fuel consumption by adapting an engine stop condition of automatically stopping an engine even during traveling has also been proposed (e.g., see PTL 1). Further, a technique to supply a hydraulic pressure to an after-shift frictional engagement element to realize a gear position after a downshift of the transmission during the coast-stop has been proposed, because it takes time to downshift the transmission and start the engine when an acceleration request is received from the driver during coast-stop (e.g., see PTL 2).